Why does the emission wavelength directly proportional to its duration?
Your question is not obvious , but in general the relation between duration and the wavelength is as follows : During a transition which produces a photon with a certain wavelength the duration of obtaining a continuous stream of photons or radiation depends on the whether the transitions happen to be within higher levels or lower levels due to different level time and the continuity of pumping . Assuming the radiation to be from dipole oscillations then we have a decay time depends on the initial energy given. High initial energy leads to long duration of radiation.
I found that the duration of solar flares tends to be much shorter than that of the radio bursts. So, I was thinking that there's a relation between the emission duration and its wavelength.
Solar flares and type II radio bursts have different origins and thus their durations should not be similar. It is believed that solar flares are caused by release of energy and plasma heating/acceleration due to sudden disruption of magnetic structures of parent active regions. Type II bursts are related to shock waves which can be generated in the low corona and can propagate far away in the interplanetary medium that can lasts dozens of hours. Life time of shock waves is, in general, independent of duration of an accompanying flare, even in the case when a flare is a driver of a shock wave (the case of a blast wave). In such case, a flare just generate a blast shock impulsively, and the shock is propagating freely after that. There is another case, when a shock is driven by a coronal mass ejection (the case of a piston shock). In such case, the shock wave and an associated type II burst is not dependant on an accompanying flare at all.
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